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We say that a category is algebraically complete when every endofunctor has an initial algebra. Similarly, a category is algebraically cocomplete when every endofunctor has a final coalgebra.

Assuming the Generalized Continuum Hypothesis, the category of sets of cardinality less or equal a certain regular uncountable ordinal, $\mathsf{Set}_\lambda$, is algebraically complete and cocomplete. This is proven, for instance, in Adamek's "On Terminal Coalgebras Derived from Initial Algebras".

What about categories of presheaves valued in $\mathsf{Set}_\lambda$? Is $[\mathbb{C},\mathsf{Set}_\lambda]$, for $\mathbb{C}$ with cardinality of objects and morphisms less or equal than $\lambda$, algebraically complete and cocomplete? Is there any reference for the topic of when do (terminal)initial (co)algebras for endofunctors over presheaf categories exist?


When $\mathbb{C}$ is a discrete category, the same paper explains that $[\mathbb{C},\mathsf{Set}_\lambda]$ is algebraically complete and cocomplete whenever $\lambda > |\mathbb{C}|$. This uses the fact that every object in the category can be written as the coproduct of the functors choosing 1 at some object and 0 everywhere else. However, it seems to me that the same argument cannot be applied when $\mathbb{C}$ is not discrete: for instance, there are many presheaves over the diagram with two parallel arrows that cannot be decomposed as the coproduct of arrows.


I have also found a result saying that accessible endofunctors on locally presentable categories that preserve monics do have a terminal coalgebra (see, for instance, Worrell's "Terminal sequences for accessible endofunctors"). Can we have a stronger result if we restrict to presheaf categories?

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    $\begingroup$ In the definition of an algebraically co/complete category, do you assume that your endofunctors are accessible? I suspect that every presheaf category over a nonempty category admits a non-accessible endofunctor, and I doubt that such endofunctors have initial algebras / terminal coalgebras. I also find the terminology unfortunate -- I'd probably prefer to use "cocomplete" for the initial algebras (which are more of a colimit-type construction) and "complete" for the final coalgebras (which are more of a limit-type construction). $\endgroup$
    – Tim Campion
    Dec 28, 2021 at 23:02
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    $\begingroup$ The ordinary meaning of "category of presheaves over $C$" is a category of the form $[C^{\rm op},\rm Set]$ for a small category $C$, and such categories are never algebraically (co)complete, because they are toposes and hence have a double-power-object functor that has no fixed points. However, it looks like Mario is interested in categories of presheaves valued in $\rm Set_{\le \lambda}$ rather than over $\rm Set_{\le\lambda}$, in which case the answer is less clear. $\endgroup$ Jan 2, 2022 at 6:18
  • $\begingroup$ I was also confused upon first reading the question because normally I would read "sets below a regular uncountable ordinal" as meaning sets of cardinality strictly less than $\lambda$, and in this case it would include the category of sets below some inaccessible cardinal, which is an (elementary) topos and hence has endofunctors with no fixed points. But upon following the link to Adamek's paper I see that he actually considers categories of sets with cardinality less than or equal to $\lambda$. $\endgroup$ Jan 2, 2022 at 6:21
  • $\begingroup$ Thank you, I see how this was confusing. I changed the wording of the question. $\endgroup$ Jan 3, 2022 at 16:23
  • $\begingroup$ FWIW: When $\lambda = 1$, $[C, Set_\lambda]$ is a complete lattice, so it is algebraically complete and algebraically cocomplete by the Knaster-Tarski theorem (no (G)CH needed). Is there any particular reason you're interested in categories of the form $[C, Set_\lambda]$? Personally, I've rarely had reason to think about categories of this form in the past... $\endgroup$
    – Tim Campion
    Jan 3, 2022 at 19:55

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